reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem LMMAZU:
  for S, T be RealNormSpace,
     f be LinearOperator of S, T holds
   f is isometric
      iff
    for x being Element of S holds ||. f.x .|| = ||. x .||
  proof
    let S, T be RealNormSpace,
        f be LinearOperator of S, T;
   hereby assume
  A1: f is isometric;
   thus for x being Element of S holds ||. f.x .|| = ||. x .||
  proof
    let x be Element of S;
    thus ||. f.x .|| = ||. f.(x -0.S) .|| by RLVECT_1:13
                 .= ||. f.x - f. (0.S) .|| by LM001
                 .= ||. x  -0.S .|| by A1
                 .= ||. x .|| by RLVECT_1:13;
   end;
 end;
  assume A2: for x being Element of S holds ||. f.x .|| = ||. x .||;
  for a,b being Point of S holds ||. f.a - f.b .|| = ||. a - b .||
   proof
    let a,b be Point of S;
    thus ||. f.a - f.b .|| = ||. f.(a -b) .|| by LM001
                        .= ||. a - b .|| by A2;
   end;
  hence thesis;
 end;
